A while ago (a long while ago, I guess — hey, I have a job, so I’m busy), I posted about how 19-tone equal temperament is a diatonic tuning. But is it the *only* one? Other than 12-TET, that is?

Obviously not. Let’s take a look at how it works!

Under our harmonic system, we base everything on the fifth. A perfect fifth is a frequency ratio of exactly 1.5, which translates to about 701.955 cents. Our 12-TET scale only uses multiples of 100 cents, so it approximates the perfect fifth quite well at 700 cents. This is far from the only possible value, however! There’s a small range where we can arbitrarily choose our fifth, and that’s what we’re about to explore.